We set, for \(U\)  a connected component of \(\mathcal {J}_n\) :
\((u_n)_U&:=\frac{1}{\#( \partial U)}\sum _{J\in \partial U} u_n(J).\) 

We then set
\(Tu_n(J)=\left\lbrace  \begin{array}{ll}u_n(J) & \ \ \text{if $J\in \mathcal {I}_{s_n}(Q)\setminus \mathcal {J}_{n}$},\\(u_n)_U \ & \ \text{if $J\in U$, for some $U$ connected component of $ \mathcal {J}_n$}.\end{array}\right.\) 
We now estimate the energy of \(Tu_n\) .
\(\sum _{\begin{array}{c}J,J^{\prime }\in \mathcal {I}_{s_n}(Q) \\ |J-J^{\prime }|=s_n\end{array}} |Tu_{n}(J)-Tu_{n}(J^{\prime })|^q s_n^{d-q} \le & C \sum _{\begin{array}{c}J,J^{\prime }\in \mathcal {I}_{s_n}(Q)\setminus \mathcal {J}_n \\ |J-J^{\prime }|=s_n\end{array}} |u_n(J)-u_n(J^{\prime })|^q s_n^{d-q} \nonumber \\&+C\sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} }\sum _{\begin{array}{c}J\in \partial U\end{array}} |u_n(J)-(u_n)_U|^q s_n^{d-q}. \) 

Let us treat the second sum separately. Fix \(U\subset \mathcal {J}_n \)  a connected component and observe, by considering the paths given by Lemma REF , that
\(\sum _{\begin{array}{c}J\in \partial U\end{array}}& |u_n(J)-(u_n)_U|^q s_n^{d-q} \le \frac{1}{\#(\partial U)} \sum _{\begin{array}{c}J,J^{\prime }\in \partial U\end{array}} |u_n(J)-u_n(J^{\prime })|^q s_n^{d-q}\\&\le \frac{1}{\#(\partial U)} \sum _{\begin{array}{c}J,J^{\prime }\in \partial U\end{array}} (\#(\mathbf {t}_{\partial U}(J,J^{\prime }))^{q-1}\sum _{I\in \mathbf {t}_{\partial U}(J,J^{\prime })}\sum _{\begin{array}{c} I^{\prime } \in \mathcal {I}_{s_n}(Q)\setminus \mathcal {J}_n :\\ I^{\prime }\in N_{\varepsilon _n}(I)\end{array} } |u_n(I)-u_n(I^{\prime })|^q s_n^{d-q} \\&\le \frac{C}{\#(\partial U)} \sum _{\begin{array}{c}J,J^{\prime }\in \partial U\end{array}} (\#(\mathbf {t}_{\partial U}(J,J^{\prime }))^{q-1} \sum _{I\in \mathbf {t}_{\partial U}(J,J^{\prime })}\left(\sum _{\begin{array}{c}I^{\prime }\in \mathcal {I}_{s_n}(Q)\setminus \mathcal {J}_n :\\ I^{\prime }\in N_{\varepsilon _n}(I)\end{array} } |u_n(I)-u_n(I^{\prime })|^p s_n^{\frac{dp}{q}-p}\right)^{q/p} \\&\le \frac{C s_n^{\frac{d(p-q)}{p}}}{\#(\partial U)} \sum _{\begin{array}{c}J,J^{\prime }\in \partial U\end{array}} (\#(\mathbf {t}_{\partial U}(J,J^{\prime }))^{q-1}\sum _{I \in \mathbf {t}_{\partial U}(J,J^{\prime })}\left(\sum _{\begin{array}{c}I^{\prime }\in \mathcal {I}_{s_n}(Q)\setminus \mathcal {J}_n :\\ I^{\prime }\in N_{\varepsilon _n}(I)\end{array} } |u_n(I)-u_n(I^{\prime })|^p s_n^{d-p}\right)^{q/p}\) 

where we applied Jensen's inequality with the function \(x\mapsto x^{p/q}\) . Call
\(Eu_n(I):= \sum _{\begin{array}{c}I^{\prime }\in \mathcal {I}_{s_n}(Q)\setminus \mathcal {J}_n :\\ I^{\prime }\in N_{\varepsilon _n}(I)\end{array} } |u_n(I)-u_n(I^{\prime })|^p s_n^{d-p}.\) 
Then, an application of Holder's inequality yields
\(\sum _{\begin{array}{c}J\in \partial U\end{array}} & |u_n(J)-(u_n)_U|^q s_n^{d-q} \le \frac{C s_n^{\frac{d(p-q)}{p}}}{\#(\partial U )} \sum _{\begin{array}{c}J,J^{\prime }\in \partial U\end{array}} (\#(\mathbf {t}_{\partial U}(J,J^{\prime }))^{q-1}\sum _{I \in \mathbf {t}_{\partial U}(J,J^{\prime })}\left(Eu_n(I)\right)^{q/p}\\&\le C \frac{s_n^{\frac{d(p-q)}{p}}}{\#(\partial U)} \sum _{\begin{array}{c}J,J^{\prime }\in \partial U\end{array}} (\#(\mathbf {t}_{\partial U}(J,J^{\prime }))^{q-1+\frac{p-q}{p}}\left(\sum _{I\in \mathbf {t}_{\partial U}(J,J^{\prime })} Eu_n(I)\right)^{q/p}\\&\le C s_n^{\frac{d(p-q)}{p}} \#(\partial U)^{q+\frac{p-q}{p}}\left(\sum _{I\in \partial U} Eu_n(I)\right)^{q/p} .\) 

In the last inequality we applied (REF ). In particular, by applying again Holder inequality
\(\sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } &\sum _{\begin{array}{c}J\in \partial U\end{array}} |u_n(J)-(u_n)_U|^q s_n^{d-q}\\& \le s_n^{\frac{d(p-q)}{p}} \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \#(\partial U)^{q+\frac{p-q}{p}}\left(\sum _{I\in \partial U} Eu_n(I)\right)^{q/p}\\& \le s_n^{\frac{d(p-q)}{p}} \left( \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \#(\partial U)^{\frac{pq}{p-q}+1}\right)^{\frac{p-q}{p}}\left( \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \sum _{I\in \partial U } Eu_n(I)\right)^{q/p}.\) 

By assumption (REF ) we have
\(\left( \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \sum _{I\in \partial U } Eu_n(I)\right)^{q/p}<M.\) 
Moreover, by Properties b), c) of \(\mathcal {J}_n\)  (Definition REF ) we can infer (since \(\#(\partial U)\le C \#(U)\)  for some universal constant \(C\) )
\(s_n^{\frac{d(p-q)}{p}} \left( \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \#(\partial U)^{\frac{pq}{p-q}+1}\right)^{\frac{p-q}{p}}&\le C\left( \log (s_n^{-1})^{\frac{pq}{p-q}} \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \#(\partial U)s_n^{d}\right)^{\frac{p-q}{p}}\\&\le C\left( \log (s_n^{-1})^{\frac{pq}{p-q}}\#(\mathcal {J}_n) s_n^{d}\right)^{\frac{p-q}{p}}\\&\le C\left( \log (s_n^{-1})^{\frac{pq}{p-q}}s_n^{\varrho _0}\right)^{\frac{p-q}{p}}.\) 

This is enough to deduce that, for \(q<p\) ,
\(\lim _{n\rightarrow +\infty }\sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \sum _{\begin{array}{c}J\in \partial U\end{array}} |u_n(J)-(u_n)_U|^q s_n^{d-q}=0\) 
which, combined with (REF ), (REF ) is enough to conclude a uniform bound
\(\sup _{n\in \mathbb {N}} \left\lbrace \sum _{\begin{array}{c}J,J^{\prime }\in \mathcal {I}_{s_n}(Q) \\ |J-J^{\prime }|=s_n\end{array}} |Tu_n(J)-Tu_n(J^{\prime })|^q s_n^{d-q}\right\rbrace  <+\infty .\) 
The \(L^q\)  norm part can be estimated in a similar way as follows.
\(\sum _{J\in \mathcal {I}_{s_n}(Q)} |Tu_n(J)|^{q} s_n^d &\le \sum _{J\in \mathcal {I}_{s_n}(Q)\setminus \mathcal {J}_n} |u_n(J)|^{q} s_n^d+\sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \frac{\#(U)}{\#(\partial U)} \sum _{J\in \partial U}|u_n(J)|^q s_n^d.\) 

Observe that, recalling the notation \(E(U):=\bigcup _{J\in U} Q_{s_{n}}(J)\) , we have (for some constant \(C\)  depending on the dimension only)
\(\begin{split}C^{-1} \#(\partial U) s_n^{d-1}\le P(&E(U))\le C \#(\partial U) s_n^{d-1}\\C^{-1} \#(\partial U) s_n^{d}\le |&E(U)| \le C \#(\partial U) s_n^{d}\end{split}\) 
(being \(P(\cdot )\)  the distributional perimeter). Thus, by the isoperimetric inequality (see for instance [1]})
\(\frac{\#(U)}{\#(\partial U)}\le C \frac{1}{s_n}\frac{|E(U)|}{P(E(U))}\le C \frac{1}{s_n} P(E(U))^{1/(d-1)}\le C \#(\partial U)^{1/(d-1)}\) 
where \(C\)  is a universal constant depending only on the dimension. Thus
\(\sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \frac{\#(U)}{\#(\partial U)} \sum _{J\in \partial U}|u_n(J)|^q s_n^d &\le C \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \#(\partial U)^{1/(d-1)} \sum _{J\in \partial U}|u_n(J)|^q s_n^d.\) 

Also
\(\sum _{J\in \partial U}|u_n(J)|^q s_n ^d &\le s_n^d \#(\partial U)^{(p-q)/p} \left(\sum _{J\in \partial U}|u_n(J)|^p \right)^{q/p} \\&\le s_n^{d(p-q)/p} \#(\partial U)^{(p-q)/p} \left(\sum _{J\in \partial U}|u_n(J)|^p s_n^d \right)^{q/p} .\) 

Call
\(\Vert u_n\Vert _{\ell ^p(\partial U)}:=\left(\sum _{J\in \partial U}|u_n(J)|^p s_n^d\right)^{1/p}\) 
hence
\(\sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } & \frac{\#(U)}{\#(\partial U)} \sum _{J\in \partial U}|u_n(J)|^q s_n ^d \le C s_n^{d(p- q)/p}\sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \#(\partial U)^{\frac{1}{d-1}+\frac{p-q}{p}} \Vert u_n\Vert _{\ell ^p(\partial U)}^q\\&\le \left( \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \#(\partial U)^{\frac{p}{(p-q)(d-1)}+1} s_n^{d} \right)^{\frac{p-q}{p}}\left( \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \Vert u_n\Vert _{\ell ^p(\partial U)}^p\right)^{q/p}.\) 

By assumption
\(\left(\sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \Vert u_n\Vert _{\ell ^p(\partial U)}^p\right)^{q/p}<M.\) 
Moreover, by properties b), c) of \(\mathcal {J}_n\)  (Definition REF ) using again that \(\#\partial U \le C \# U\)  we have
\(\left( \sum _{\begin{array}{c}U\subset \mathcal {J}_n \\ \text{connected}\\\text{component}\end{array} } \#(\partial U)^{\frac{p}{(p-q)(d-1)}+1} s_n^{d} \right)^{\frac{p-q}{p}}\le C \left(\log (s_n^{-1})^{\frac{p}{(p-q)(d-1)}} s_n^{\varrho _0} \right)^{\frac{p-q}{p}}.\) 

We conclude that
\(\lim _{n\rightarrow +\infty } \sum _{J\in \mathcal {I}_{s_n}(Q)} |Tu_n(J)|^{q} s_n^d\le C \sum _{J\in \mathcal {I}_{s_n}(Q)\setminus \mathcal {J}_n } |u_n(J)|^{p} s_n^d.\) 
In particular, by combining (REF ), (REF ) and (REF ) we conclude (REF ).
